Solve for $t$, $ -\dfrac{6}{25t + 15} = -\dfrac{4t + 10}{5t + 3} + \dfrac{5}{10t + 6} $
Solution: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $25t + 15$ $5t + 3$ and $10t + 6$ The common denominator is $50t + 30$ To get $50t + 30$ in the denominator of the first term, multiply it by $\frac{2}{2}$ $ -\dfrac{6}{25t + 15} \times \dfrac{2}{2} = -\dfrac{12}{50t + 30} $ To get $50t + 30$ in the denominator of the second term, multiply it by $\frac{10}{10}$ $ -\dfrac{4t + 10}{5t + 3} \times \dfrac{10}{10} = -\dfrac{40t + 100}{50t + 30} $ To get $50t + 30$ in the denominator of the third term, multiply it by $\frac{5}{5}$ $ \dfrac{5}{10t + 6} \times \dfrac{5}{5} = \dfrac{25}{50t + 30} $ This give us: $ -\dfrac{12}{50t + 30} = -\dfrac{40t + 100}{50t + 30} + \dfrac{25}{50t + 30} $ If we multiply both sides of the equation by $50t + 30$ , we get: $ -12 = -40t - 100 + 25$ $ -12 = -40t - 75$ $ 63 = -40t $ $ t = -\dfrac{63}{40}$